Linear property of the screw surface of the spatial RPRP linkage

This paper investigates the screw surface of the RPRP linkage, an overconstrained linkage with mobility one. The screw surface is a ruled surface containing screws for displacing the coupler of the RPRP linkage from one reference position to all reachable positions. In this paper, two types of RPRP linkages, in folded and unfolded forms, are constructed by using hexahedrons in accordance with Delassus' criteria. It is shown that the screw surfaces of both types of RPRP linkages can be represented by screw systems of the second order. These novel finite screw systems are obtained by intersecting two 3-systems corresponding to the finite displacements of the RP and PR dyads. The intersection of finite screw systems is conducted hy employing the intersection operation of vector subspaces of the six-dimensional vector space.